New Methodological Developments for the International Comparison Program
Erwin Diewert,1
Discussion Paper 08-08, Department of Economics,
The University of British Columbia, Vancouver, Canada, V6T 1Z1.
email: [email protected] Revised August 3, 2010.
Abstract
The paper explains the new methodology that was used in the 2005 International Comparison Program (ICP) that compared the relative price levels and GDP levels across 146 countries. In this round of the ICP, the world was divided into 6 regions: OECD, CIS, Africa, South America, Asia Pacific and West Asia. What is new in this round compared to previous rounds of the ICP is that each region was allowed to develop its own product list and collect prices on this list for countries in the region. The regions were then linked using another separate product list and 18 countries across the 6 regions collected prices for products on this list and this information was used to link prices and quantities across the regions. An additional complication was that the final linking of prices and volumes across regions had to respect the regional price and volume measures that were (separately) constructed by the regions. The paper also studies the properties of the Iklé Dikhanov Balk multilateral system of index numbers which was used by Africa.
Journal of Economic Literature Classification Numbers
C43, C81, E31, O57.
Keywords
Index numbers, multilateral comparison methods, GEKS, EKS, Geary-Khamis, Balk, Dikhanov, Iklé, Country Product Dummy (CPD) method, basic headings, Structured Product Descriptions, Purchasing Price Parities (PPPs), representative products, spatial chaining, fixity.
1. Introduction
The final results for the 2005 International Comparison Program (ICP) have been released in February; for a tabulation of the results, see the World Bank (2008). The
1
This is a revised version of a paper presented at the Joint UNECE/ILO Meeting on Consumer Prices Indices, May 8-9, Palais des Nations, Geneva. The author thanks Bert Balk, Yonas Biru, Yuri Dikhanov, Louis Marc Ducharme, Alan Heston, Peter Hill, Alice Nakamura, Sergey Sergeev, Fred Vogel and Kim Zieschang for helpful discussions and comments and the World Bank for financial support. None of the above are responsible for any opinions expressed by the author. For a less technical version of this paper, see Diewert (2008).
program compared the level of prices and the quantities or volumes of GDP (and its components) for 146 countries for the year 2005. International price statisticians developed Structured Product Descriptions (SPDs) for approximately 1000 products2 and the individual countries collected price information on these products for the year 2005. The 1000 products were grouped into 155 Basic Heading (BH) categories. The price information collected in each country was then compared across countries, leading to a matrix of 155 basic heading prices by 146 countries. The precise way in which the individual product prices in each BH category were aggregated into a single country price for each BH heading is the topic which will be investigated in sections 2 and 3 below. The 2005 ICP differed from previous ICP rounds.3 In previous rounds, each country attempted to find prices in their country for a common product list. However, it is difficult to find products that are representative for all countries in the world and so the decision was made to break up the world into 6 regions and price statisticians developed
separate product lists for each region. The 6 regions were: (1) Africa with 48
participating countries; (2) South America with 10 countries; (3) Asia Pacific with 23 countries; (4) The Commonwealth of Independent States (CIS) with 10 countries; (5) West Asia with 11 countries and (6) the OECD and other European countries covered by Eurostat plus Israel and Russia adding up to 46 countries in this region. This sums to 148 countries but Egypt appears in both the African and West Asia regions and Russia appears in both the OECD and CIS regions so there are 146 participating countries in all. The fact that the product lists in each region were allowed to be different across regions means that without further information, prices and volumes could not be compared across regions. However, the World Bank, in cooperation with other national and international statistical agencies, developed an additional product list, which was priced out by 18 selected countries across the regions. These 18 countries were called ring countries. The prices that were collected by the ring countries using this final product list enabled price comparisons to be made across the 6 regions. We will indicate how this was done at the Basic Heading level in section 3 below and in section 5, we will indicate how comparisons at higher levels of aggregation between regions were made.
There was another methodological innovation made in this current ICP round in addition to having regional product lists: the price parities or Purchasing Power Parities (PPPs) and relative volumes for each country were determined using information on prices and GDP expenditure shares that pertained only to countries within the given region and these parities and relative volumes were preserved in the world comparison. Thus each region was independently allowed to determine its country PPPs and volume shares and the final linking of the regional results into a global world comparison left these regional relative parities undisturbed.4
2 Most of the products referred to are components of individual consumption: “There are about 830 SPDs that cover 100 Basic Headings for individual consumption. Each SPD contains price determining characteristics that will define unique products from any corner of the world.” Dennis Trewin (2008; 8). For an overview of the organization and methodology used in the 2005 ICP, see Vogel (2008).
3 For an overview of previous ICP rounds and an assessment of the current round, see Heston and Summers (2008).
The final results from the 2005 International Comparison Program for the 146 participating countries are available on the World Bank website; see the World Bank (2008) for these results and explanations for various difficulties that were encountered. This publication explained the basic framework for the provision of the data as follows: “The purchasing power parities and the derived indicators in this report are the product of a joint effort by national statistical offices, regional coordinators, and the ICP global office. PPPs cannot be computed in isolation by a single country. However, each country was responsible for submitting official estimates of 2005 gross domestic product and its components, population counts, and average exchange rates. The regional coordinators worked with the national statistical offices to review the national accounts data to ensure that they conformed to the standards of the 1993 System of National Accounts. Similar reviews were conducted for population and exchange rate data.” The World Bank (2008; 2)
The World Bank noted that the data provided by China were not quite complete and that the Tables broke China into 4 separate regions:
“China submitted prices for 11 administrative areas and the urban and rural components. The World Bank and the Asian Development Bank extrapolated these 11 city prices to the national level. The China data do not include Hong Kong, Macao, and Taiwan, China.” The World Bank (2008; 2).
The World Bank publication also explained how the ICP dealt with the fact that Egypt appeared in two regions (and priced out the product lists for both regions):
“Egypt participated in both the Africa and West Asia ICP programs by providing prices for the products included in each comparison. Therefore, it was possible to compute PPPs for Egypt separately for Africa and West Asia. Both regions included Egypt results in their regional reports. Egypt appears in the global report in both regions. The results for Egypt from each region were averaged by taking the geometric mean of the PPPs, allowing Egypt to be shown in each region with the same ranking in the world comparison.” The World Bank (2008; 2).
Finally, the World Bank explained how the CIS regional results were obtained:
“Russia participated in the price collection for both the CIS and OECD comparisons. As with Egypt, PPPs for Russia were computed separately for the OECD and CIS comparisons. However, the CIS region did not participate in the Ring. Therefore, following past practices the CIS region was linked to Eurostat-OECD using Russia as a link. For comparison purposes, Russia is shown in both regions in the report.” The World Bank (2008; 2).
Thus since Russia is the only country that belongs to both the OECD region and the CIS region, linking the two regions at both the Basic Heading level and higher levels of aggregation can be done though Russia. The same linking strategy could have been used to link the Africa and West Asia regions using Egypt as the linking country (or bridge country using ICP parlance) but a decision was made not to do this.5
5 The problems in the case of Egypt are more complicated than in the case of Russia since there were more than one ring countries in Africa and in West Asia. Hill (2007c; 13) listed the 18 ring countries as Brazil, Cameroon, Chile, Egypt, Estonia, Hong Kong, Japan, Jordan, Kenya, Malaysia, Oman, Philippines, Senegal, Slovenia, South Africa, Sri Lanka, United Kingdom and Zambia. Thus Cameroon, Jordan, Kenya, Oman, Senegal, South Africa and Zambia join Egypt as ring countries that are present in either the African or West Asian regions.
The above material presents a quick overview of the ICP. Our specific task in the present paper is to present some of the methodological details of the methods that were used to: • Link the Basic Heading PPPs across the regions (sections 2 and 3) and
• Link the price levels and volumes for each country within a region across the regions in a way that preserves the regional relative price and volume measures (sections 4 and 5).
Thus sections 2 and 3 deal with the problems associated with the aggregation of price information at the lowest level of aggregation where information on expenditures or quantities is not available. Sections 4 and 5 deal with aggregation problems at higher levels of aggregation where expenditure information by category and country is available. It should be noted that the material to be covered in sections 2-5 below overlaps substantially with the material in the ICP 2003-2006 Handbook; see Hill (2007a) (2007b) (2007c) (2007d) (2007e). Also the material in sections 2 and 3 overlaps with Hill (2008) and the material in sections 3 and 5 overlaps substantially with Diewert (2004b).
Section 6 lists some of the methodological problems that require additional research before the next round of the ICP program, which is scheduled to take place in 2011. Section 7 provides some concluding remarks to the main text. An Appendix, which looks at the properties of the relatively new Iklé Dikhanov Balk multilateral system used by the African region, concludes the paper.
2. The Comparison of Prices Across Countries Within a Region at the BH Level
Three distinct methods for linking prices across countries within a region at the Basic Heading level were used by the regions in the 2005 ICP:
• The Country Product Dummy (CPD) method (used by the African, Asian Pacific and West Asian regions);
• The Extended Country Product Dummy (CPRD) method (used by South America) and
• The EKS* method used by the OECD/Eurostat and CIS regions.
The most widely used statistical approach to the multilateral aggregation of prices at the first stage of aggregation is the Country Product Dummy (CPD) method, proposed by Robert Summers (1973). This method for making international comparisons of prices can be viewed as a very simple type of hedonic regression model where the only characteristic of the commodity is the commodity itself. The CPD method can also be viewed as an example of the stochastic approach6 to index numbers. Since an extension
6 See Selvanathan and Rao (1994) for examples of the stochastic approach to index number theory. A main advantage of the CPD method for comparing prices across countries over traditional index number methods is that we can obtain standard errors for the country price levels. This advantage of the stochastic approach to index number theory was stressed by Summers (1973).
of this method was used to link prices across regions, we will outline the algebra behind this approach.
Suppose that we are attempting to make an international comparison of prices between C countries over a reasonably homogeneous group of say N items.7 In this section, we also assume that no expenditure weights are available for the price comparisons. Let pcn
denote the average price of item n8 in country c for c = 1,…,C; n = 1,…,N. Each item n must be measured in the same quantity units across countries but the prices are collected in local currency units. The basic statistical model that is assumed is the following one: (1) pcn = acbnecn ; c = 1,…,C; n = 1,…,N.
where the ac and bn are unknown parameters to be estimated and the ecn are independently
distributed error terms with means 1 and constant variances. The parameter ac is to be
interpreted as the average level of prices (over all items in this group of items) in country c relative to other countries and the parameter bn is to be interpreted as the average (over
all countries) multiplicative premium that item n is worth relative to an average item in this grouping of items. Thus the ac are the basic heading country price levels that we
want to determine while the bn are item or individual product effects. The basic
hypothesis is that the price of item n in country c is equal to a country price level ac times
an item commodity adjustment factor bn times a random error that fluctuates around 1.
Taking logarithms of both sides of (1) leads to the following model:
(2) ycn = αc + βn + εcn ; c = 1,…,C; n = 1,…,N
where ycn ≡ ln pcn, αc ≡ ln ac, βn ≡ ln bn and εcn ≡ ln ecn.
The model defined by (2) is obviously a linear regression model where the independent variables are dummy variables. The least squares estimators for the αc and βn can be
obtained by solving the following minimization problem:9
(3) min α’s, β’s {∑c=1 C ∑
n=1N [ycn − αc − βn]2}.
However, it can be seen that the solution for the minimization problem (3) cannot be unique: if αc* for c = 1,…,C and βn* for n = 1,…,N solve (3), then so does αc* + γ for c =
1,…,C and βn* − γ for n = 1,…,N, for any arbitrary number γ. Thus it will be necessary to
impose an additional restriction or normalization on the parameters αc and βn in order to
7 Using the language of the International Comparison of Prices (ICP) project, we are making a comparison of prices at the basic heading level. In ICP 2005 project, there are 155 basic headings. Thus each region using this method would have to run 155 regressions of the type described here.
8 In most cases, this item n price in country c was an unweighted arithmetic mean of prices collected over outlets and regions in the country during the reference year.
9 Weighted (by expenditure shares) versions of the CPD model were considered by Prasada Rao (1990), (1995) (2001) (2002) (2004), Heston, Summers and Aten (2001), Sergueev (2002) (2003), Diewert (2004b) (2005), Hill (2007a; 23-24) and Hill and Timmer (2006).
obtain a unique solution to the least squares minimization problem (3). The simplest normalization is:
(4) α1 = 0 or a1 = 1 ;
The normalization (4) means that country 1 is chosen as the numeraire country and the parameter ac for c = 2,…,C is the PPP (Purchasing Power Parity) of country c relative to
country 1 for the class of commodity prices that are being compared across the C countries.10
Cuthbert and Cuthbert (1988; 57) introduced an interesting generalization of the Country Product Dummy method that can be used if information on representativity of the prices is collected by the countries in the comparison project along with the prices themselves. Hill (2007a) (2008) explains this method in some detail and he called the method the
extended CPD Method or CPRD Method and he justified the method as follows:
“The reason for distinguishing between representative and unrepresentative products is that the relative prices of representative products in a country may be expected to be low compared with relative prices of the same products in countries in which they are not representative. Conversely, of course, the relative prices of unrepresentative products will tend to be high. This will tend to happen as result of normal substitution effects. Products will tend to be purchased in relatively large (small) quantities precisely because their relative prices are low (high). This conclusion is not merely a theoretical deduction, as there is ample empirical evidence of the substitution effect at work in both inter-temporal and inter-national comparisons.” Peter Hill (2007a; 3).
“The expected price depends on the interaction of three factors: the country, the product and its representativity. Given that the coefficient of a representative product is fixed at unity, the coefficient of an unrepresentative product may be expected to be greater than unity. The price of product is expected to be higher relatively to the reference product 1 in a country in which it is unrepresentative than in a country in which it is representative. The improvement over the traditional CPD method comes from the partial relaxation of the unrealistic assumption that the pattern of relative prices is the same in all countries. ... The addition of the new variable, representativity, does not simply add another parameter to be estimated. It adds another dimension to the analysis. As there are three types of explanatory variables in the regression -- country, product and representativity -- the extended regression will described as the CPRD method to distinguish it from the traditional CPD method.” Peter Hill (2007a; 26).
The basic idea is that representative products in a country should tend to be lower in price (and hence they should be more popular) compared to unrepresentative products; thus representativity becomes a price determining characteristic of the commodity.
The CPDR method generalizes the model (2) above as follows. Define ycnu = ln pcnu
where pcnu is the logarithm of the average product n price collected in country c and u is
an index that denotes whether the collected price is unrepresentative (in which case u = 1) or representative (in which case u = 2). The basic (unweighted) statistical model that is assumed is the following one:
(5) ycnu = αc + βn + δu + εcnu ; c = 1,…,C; n = 1,…,N; u = 1,2
10 See Rao (2004) and Hill (2007a) for further analysis of this model. We note that Hill uses a different but equivalent normalization.
where the αc are the log country PPP’s, the βn are the log product price effects and the δu
are the two log representativity effects and the εcnu are independently distributed random
variables with mean zero and constant variances. In order to identify the parameters, the following normalizations can be used:
(6) α1 = 0 ; δ1 = 0.
Thus the present model is much the same as the basic CPD model except that we have 3 classifications instead of 2. For additional discussion of this model, the reader is referred to Cuthbert and Cuthbert (1988), Cuthbert (2000), Diewert (2004b) and Hill (2007a) (2008).
We agree with Hill in endorsing the method in theory. However, in practice, it seems it was at times difficult for national price statisticians to agree on a workable definition of representativity that was uniform across countries and regions. Thus in the end, it appears that only the South American region used CPRD method to construct its 155 by 10 matrix of PPP’s by Basic Heading and country. The other regions used the basic CPD method or the EKS* method (which also used the representativity concept).
As mentioned at the beginning of this section, the EKS* method was used to aggregate prices at the lowest level of aggregation in the OECD and CIS regions. This method is explained by Hill as follows:
“Eurostat abandoned EKS 1 in 1982 and replaced it by the method described in the present section, which will be called the asterisk method or EKS*. A detailed exposition of EKS* and its properties is given by Sergey Sergeev (2003). The EKS* method is so called because it makes use of the distinction between representative and unrepresentative products, the representative products being identified in the product lists by an *. The EKS* method recognizes, and exploits, the fact that, as already explained, the prices of representative products are likely to be relatively low, whereas the prices of unrepresentative products are likely to be relatively high. The method proceeds by calculating two separate Jevons indices for each pair of countries. One Jevons index covers products that are representative in the first country, treated here as the base country. The other covers products that are representative in the second country. Of course, some products may be representative in both countries and included in both indices. The two indices may be described as Jevons 1 and Jevons 2 respectively.” Peter Hill (2007a; 9).
Thus two bilateral Jevons type indexes are calculated for any two countries. Jevons 1 (2) compares only the price relatives of products that are representative in country 1 (2). The final bilateral index of prices between the two countries under consideration is a geometric mean of the two Jevons indexes.11 Once all of these bilateral parities have
been constructed over each pair of countries in the region, they can be harmonized by
11 Note that prices which are not representative in both countries but are collected in both countries do not appear in the final bilateral index of prices between the two countries. This means that the EKS* procedure is not fully efficient in a statistical sense, whereas the CPRD procedure is fully efficient.
using the EKS procedure.12 For further details of this method, the reader is referred to Hill (2007a).
A majority of the members of the Technical Advisory Group who provided advice to ICP 2005 favoured the CPRD method described in the previous section over the EKS* method described in this section for two reasons:
• The CPRD method used all of the available price information whereas EKS* did not and
• The CPRD method gave straightforward measures of the statistical precision of the estimated parities.
However, it appears that Eurostat price statisticians are locked into the EKS* method by legislation and thus the OECD/Eurostat region stuck by its EKS* method in the current European Comparison Program. More research is required in order to determine how much difference there would be between CPRD and EKS*. But without having this research in hand, I would certainly favor the use of CPRD over EKS*, mainly because the EKS* method throws away valuable information on some prices and this cannot be a statistically efficient procedure.
There is also the issue of choosing between the original CPD method and the enhanced CPRD method, which makes use of representativity information on the item prices. Hill (2007a) explains theoretically why the CPRD method should be preferred over the CPD method. However, in practice, national price collectors in all of the non OECD regions had great difficulty in deciding on which items were representative and which items were not. Thus when the CPRD regressions were run, the coefficients for the representative dummy variables had more or less random signs instead of the expected signs. This was the case even for the South American region, which used the CPRD method.13 Thus at this stage of our knowledge of the various methods used to aggregate prices at the basic heading level, I would favor the use of the plain vanilla CPD method.
Having described the methods used to construct PPPs for the 155 basic headings for each country in a region, we now consider how to link these PPPs across regions.
3. The Comparison of Prices Across Regions at the Basic Heading Level
As noted in the introduction, a group of ring countries collected prices from a common list and this price information was used to link the regional basic heading prices across the 6 regions. However, since the CIS region was locked into the OECD/Eurostat region, in practice, there were only 5 regions to link, with the CIS, OECD and Eurostat countries forming a single region.
12 The EKS method is explained in more detail by Balk (1996), Diewert (1999) and Hill (2007a) (2008). The method is due to Gini (1924) (1931) and independently rediscovered by Eltetö and Köves (1964) and Szulc(1964).
The methodology used to link basic heading prices across regions was developed by Diewert (2004b; 36-39) and we review that methodology here.14 The model is basically an adaptation of the unweighted CPD model presented in section 2.1.
In order to set the stage for what was actually done in linking the regions, we first generalize the CPD model presented in section 2 to allow for a reorganization of the list of C countries into 5 regions and C(r) ring countries in each region r. Thus C(r) is not the total number of countries in region r; it is only the number of ring countries in each region because only the ring countries collected data on prices from a common international product list. With these changes, the basic model becomes:
(7) prcn ≈ ar brc cn ; r = 1,…,5; c = 1,....,C(r); n = 1,...,N;
(8) a1 = 1;
(9) br1 =1; r = 1,…,5.
The normalization (8) means that we have to choose a numeraire region. The normalizations (9) mean that within each region, we need to choose a numeraire country in order to identify all of the parameters uniquely. Thus the parameters ar and brc replace
our initial model parameters ac. Note that the total number of parameters remains
unchanged when we group all of the countries in the comparison into regions and countries within the regions.
Taking logarithms of both sides of (7) and then adding error terms εrcn (with means 0)
leads to the following regression model:
(10) ln prcn = ln ar + ln br c+ ln cn + εrcn ; r = 1,…,5; c = 1,....,C(r); n = 1,...,N;
= αr + βrc + γn + εrcnk
where we impose the following normalizations on the parameters in order to uniquely identify them:
(11) α1= 0 ;
(12) βr1 = 0 ; r = 1,…,5
where αr ≡ ln ar, βrc ≡ ln brc, γn ≡ ln cn.
If all of the data collected for each regional comparison could be pooled and if there are product overlaps between the regions, then there will be 155 regressions of the form (10) to run, one for each basic heading category. In the above model, the interregional log parities (the αr) are estimated along with the within region country log parities (the βrc)
and the product log price premiums (the γn). Call this the first approach to estimating the
regional parities for each basic heading. It uses all of the available information in making comparisons between all of the countries.
14 The basic methodology is also described in Hill (2007d). However, Hill uses somewhat different normalizations than (11) and (12).
However, the above one big regression approach (for each basic heading) is not
consistent with approaches that used only the regional data to determine the within region
parities, the βrc parameters, holding r fixed. But a principle of the current ICP
methodology was that regions should be allowed to determine their own parities, independently of other regions. However, the regression model (10) can be modified to deal with this problem. If the regional log parities βrc are known, then the term βrc (which
is equal to ln brc) can be subtracted from both sides of (10), leading to the following
regression model:
(13) ln prcn − ln brc = ln ar + ln cn + εrcn ; r = 1,…,5; c = 1,....,C(r); n = 1,...,N;
or (14) ln [prcn/brc] = αr + γn + εrcn ;
where the normalization (8) still holds. Thus if the within region parities are known, then prices in each region prcn can be divided by the appropriate regional parity for that
country in that region brc, and these regionally adjusted prices can be used as inputs into
the usual CPD model that has now only the regional log parities αr and the commodity
adjustment factors γn as unknown parameters to be estimated. Call the model defined by
(11) and (14) the second approach to estimating the regional parities for each basic heading. This second approach respects the within region parities that have been constructed by the regional price administrators. It is this second approach that was used in ICP 2005.15
We now turn our attention to the problems associated with aggregating up the basic heading PPP information (along with country expenditure information) in order to form aggregate country price and volume comparisons within a region.
4. Aggregate Price and Volume Comparisons Across Countries Within a Region
Once the 155 BH price parities for each of the K countries in a region have been constructed, aggregate measures of country prices and relative volumes can be constructed using a wide variety of multilateral comparison methods that have been suggested over the years. These aggregate comparisons assume that in addition to BH price parities for each country, national statisticians have provided country expenditures (in their home currencies) for each of the 155 BH categories for the reference year 2005. Then the 155 by K matrices of Basic Heading price parities and country expenditures are used to form average price levels across all commodities and relative volume shares for each country.
There are a large number of methods that can be used to construct these aggregate Purchasing Power Parities and relative country volumes and Hill (2007b) surveys the
main methods that have been used in previous rounds of the ICP and other methods that might be used.16 Basically, only two multilateral methods have been used in previous rounds:
• The Gini-EKS (GEKS) method based on Fisher (1922) bilateral indexes and • The Geary (1958) Khamis (1972) (GK) method, which is an additive method. In the present ICP round, aggregate PPPs and relative country volumes for countries within each region were constructed for five of the six regions using the Gini-EKS method. However, the African region wanted to use an additive method and so this region used a relatively new additive method, the Iklé Dikhanov Balk (IDB) method, for constructing PPPs and relative volumes within the region.17 These methods will be discussed in more detail below. However, at this point, it may be appropriate to comment briefly on the relative merits of the GEKS, GK and IDB methods. The GK and IDB methods are additive methods; i.e., the real output of each country can be expressed as a sum of the country’s individual outputs but each output is weighted by an international price which is constant across countries. This feature of an additive method is tremendously convenient for users and so for many purposes, it is useful to have available a set of additive international comparisons.
4.1 The Gini EKS Method
It will be useful to introduce some notation at this point. Let N equal 155 and let K be the number of countries in the regional comparison for the reference year. Denote the regional PPP for country k and commodity category n by pnk > 0 and the corresponding
expenditure (in local currency units) on commodity class n by country k in the reference year by enk for n = 1,...,N and k = 1,...,K. Given this information, we can define implicit quantity levels ynk for each Basic Heading category n and for each country k as the
category expenditure deflated by the corresponding commodity PPP for that country: (15) ynk ≡ enk/pnk ; n = 1,...,N ; k = 1,...,K.
It will be useful to define country commodity expenditure shares snk as follows:
(16) snk ≡ enk/∑i=1N eik ; n = 1,...,N ; k = 1,...,K.
16 For additional methods, see Balk (1996), R.J. Hill (1997) (1999a) (1999b) (2001) (2004) and Diewert (1999).
17 Iklé (1972; 203) proposed the equations for the method in a rather difficult to interpret manner and provided a proof for the existence of a solution for the case of two countries. Dikhanov (1994; 6-9) used the much more transparent equations (16) and (18), explained the advantages of the method over the GK method and illustrated the method with an extensive set of computations. Balk (1996; 207-208) used the Dikhanov equations and provided a proof of the existence of a solution to the sytem for an arbitrary number of countries. Van Ijzeren (1983; 42) also used Iklé’s equations and provided an existence proof for the case of two countries.
Now define country vectors of BH prices as pk ≡ [p1k,...,pNk], country vectors of BH quantities as yk ≡ [y1k,...,yNk], country expenditure vectors as ek ≡ [e1k,...,eNk] and country expenditure share vectors as sk ≡ [s1k,...,sNk] for k = 1,...,K.
In order to define the GEKS parities P1,P2,...,PK, we first need to define the Fisher (1922)
ideal bilateral price index PF between country j relative to k:18
(17) PF(pk,pj,yk,yj) ≡ [pj⋅yj pj⋅yk/pk⋅yj pk⋅yk]1/2 ; j = 1,...,K ; k = 1,...,K.
The aggregate PPP for country j, Pj, is defined as follows:
(18) Pj ≡ ∏k=1K [PF(pk,pj,yk,yj)]1/K ; j = 1,...,K.
Once the GEKS Pj’s have been defined by (18), the corresponding GEKS country real
outputs or volumes Yj can be defined as the country expenditures pj⋅yj in the reference year divided by the corresponding GEKS purchasing power parity Pj:
(19) Yj ≡ pj⋅yj/Pj ; j = 1,...,K.
If we divide all of the Pj defined by (18) by a positive number, α say, then we can multiply all of the Yj defined by (19) by this same α without materially changing the GEKS multilateral method. If country 1 is chosen as the numeraire country in the region, then we set α equal to P1 defined by (18) for j = 1 and then the price level Pj is interpreted
as the number of units of country j’s currency it takes to purchase 1 unit of country 1’s currency and get an equivalent amount of utility and the rescaled Yj is interpreted as the volume of output of country j in the currency units of country 1.
It is also possible to normalize the outputs of each country in common units (the Yk) by
dividing each Yk by the sum ∑j=1K Yj in order to express each country’s real output as a
fraction or share of total regional output; i.e., we can define the country k’s share of regional output, Sk, as follows:19
(20) Sk ≡ Yk/∑j=1K Yj ; k = 1,...,K.
Of course, the country shares of regional real output, the Sk, remain unchanged after rescaling the PPPs by the scalar α.
This completes our brief overview of the Gini EKS method for making multilateral comparisons.20
18 Notation: p⋅y ≡ ∑n=1N pnyn denotes the inner product between the vectors p and y.
19 There are several additional ways of expressing the GEKS PPP’s and relative volumes; see Balk (1996) and Diewert (1999; 34-37).
20 It should be noted that all of the multilateral methods that are described in this section can be applied to subaggregates of the 155 basic heading categories; i.e., instead of working out aggregate price and volume comparisons across all 155 commodity classifications, we could just choose to include the food categories
4.2 The Geary Khamis Method
The method was suggested by Geary (1958) and Khamis (1972) showed that the equations that define the method have a positive solution under certain conditions.
The GK system of equations involves K country price levels or PPPs, P1,...,PK, and N international commodity reference prices, π1,...,πN. The equations which determine these
unknowns (up to a scalar multiple) are the following ones:
(21) πn = ∑k=1K [ynk/∑j=1K ynj][pnk/Pk] ; n = 1,...,N ;
(22) Pk = pk⋅yk/π⋅yk ; k = 1,...,K
where π ≡ [π1,...,πN] is the vector of GK regional average reference prices. It can be seen
that if we have a solution to equations (21) and (22), then if we multiply all of the country parities Pk by a positive scalar λ say and divide all of the reference prices πn by the same
λ, then we obtain another solution to (21) and (22). Hence, the πn and Pk are only
determined up to a scalar multiple and we require an additional normalization such as (23) P1 = 1
in order to uniquely determine the parities. It can also be shown that only N + K − 1 of the N equations in (21) and (22) are independent. Once the parities Pk have been
determined, the real output for country k, Yk, can be defined as country k’s nominal value
of output in domestic currency units, pk⋅yk, divided by its PPP, Pk; i.e., we have
(24) Yk = pk⋅yk/Pk ; k = 1,...,K = π⋅yk using (22). Finally, if we substitute equations (24) into the regional share equations (20), we find that country k’s share of regional output is
(25) Sk = π⋅yk/π⋅y k = 1,...,K where the region’s total output vector y is defined as the sum of the country output vectors; i.e., we have
(26) y ≡ ∑j=1K yj .
Equations (24) show how convenient it is to have an additive multilateral comparison method: when country outputs are valued at the international reference prices, values are additive across both countries and commodities. However, additive multilateral methods are not really consistent with economic comparisons of utility across countries if the in our list of N categories and use the multilateral method to compare aggregate food consumption across the countries in the region.
number of countries in the comparison is greater than two; see Diewert (1999; 48-50) and the Appendix on this point.21 In addition, looking at equations (41), it can be seen that large countries will have a larger contribution to the determination of the international prices πn and thus these international prices will be much more representative for the
largest countries in the comparison as compared to the smaller ones.22 This leads us to the next method for making multilateral comparisons: an additive method that does not suffer from this problem of big countries having undue influence in the comparison.
4.3 The Iklé Dikhanov Balk Method
Iklé (1972; 202-204) suggested the method in a very indirect way, Dikhanov (1994) (1997) suggested the much clearer system (27)-(28) below and Balk (1996; 207-208) provided the first existence proof. Dikhanov’s (1994; 9-12) equations that are the counterparts to the GK equations (21) and (22) are the following ones:
(27) πn = [∑k=1K snk [pnk/Pk]−1/∑j=1K snj]−1 ; n = 1,...,N
(28) Pk = [∑n=1N snk [pnk/πn]−1]−1 k = 1,...,K.
As in the GK method, equations (27) and (28) involve the K country price levels or PPPs, P1,...,PK, and N international commodity reference prices, π
1,...,πN. Equations (27) tell us
that the nth international price, πn, is a share weighted harmonic mean of the country k prices for commodity n, pnk, deflated by country k’s PPP, Pk. The country k share
weights for commodity n, snk, do not sum (over countries k) to unity but when we divide
snk by ∑j=1K snj, the resulting normalized shares do sum (over countries k) to unity. Thus
equations (27) are similar to the GK equations (21), except that now a harmonic mean of the deflated commodity n prices, pnk/Pk, is used in place of the old arithmetic mean and in
the GK equations, country k’s share of commodity n in the region, ynk/∑j=1K ynj, was used
as a weighting factor (and hence large countries had a large influence in forming these weights) but now the weights involve country expenditure shares and so each country in the region has an equal influence in forming the weighted average. Equations (28) tell us that Pk, the PPP for country k, Pk, is equal to a weighted harmonic mean of the country k
commodity prices, pnk, deflated by the international price for commodity n, πn, where we
sum over commodities n instead of over countries k as in equations (27). The share weights in the harmonic means defined by (28), the snk, of course sum to one when we
21 “Figure 1.1 also illustrates the Gerschenkron effect: in the consumer theory context, countries whose price vectors are far from the ‘international’ or world average prices used in an additive method will have quantity shares that are biased upward. ... It can be seen that these biases are simply quantity index counterparts to the usual substitution biases encountered in the theory of the consumer price index. However, the biases will usually be much larger in the multilateral context than in the intertemporal context since relative prices and quantities will be much more variable in the former context. ... The bottom line on the discussion presented above is that the quest for an additive multilateral method with good economic properties (i.e., a lack of substitution bias) is a doomed venture: nonlinear preferences and production functions cannot be adequately approximated by linear functions. Put another way, if technology and preferences were always linear, there would be no index number problem and hundreds of papers and monographs on the subject would be superfluous!” W. Erwin Diewert (1999; 50).
sum over n, so there is no need to normalize these weights as was the case for equations (27).
It can be seen that if we have a solution to equations (27) and (28), then if we multiply all of the country parities Pk by a positive scalar λ say and divide all of the reference prices πn by the same λ, then we obtain another solution to (27) and (28). Hence, the πn and Pk
are only determined up to a scalar multiple and we require an additional normalization such as (23).
Although the IDB equations (28) do not appear to be related very closely to the corresponding GK equations (22), it can be shown that these two sets of equation are actually the same system. To see this, note that the country k expenditure share for commodity n, snk, has the following representation:
(29) snk = pnkynk/pk⋅yk ; n = 1,...,N ; k = 1,...,K.
Now substitute equations (29) into equations (28) to obtain the following equations: (30) Pk = 1/∑n=1N snk [pnk/πn]−1 k = 1,...,K
= 1/∑n=1N [pnkynk/pk⋅yk][πn/pnk]
= pk⋅yk/∑n=1N πnynk
= pk⋅yk/π⋅yk.
Thus equations (28) are equivalent to equations (22) and the IDB system is an additive system; i.e., equations (24)-(26) can be applied to the present method just as they were applied to the GK method for making international comparisons.
In the Appendix, we will obtain many different ways of representing the IDB system of parities and we will also establish fairly weak conditions for the existence and uniqueness of the IDB parities. We will also indicate how solutions to the equations can be found. As was mentioned in the introduction, the Iklé Dikhanov Balk method was used by the African region in order to construct regional aggregates. Basically, this method appears to be an improvement over the GK method in that large countries no longer have a dominant influence on the determination of the international reference prices πn and so if
an additive method is required with more democratic reference prices, IDB appears to be “better” than GK. However, again, we caution the reader that additive multilateral methods will not generate very accurate relative volumes (from the viewpoint of the economic approach) if the number of countries is greater than three and there is heterogeneity in relative prices and quantities; see Diewert (1999; 50) and the final section in the Appendix.
We now turn our attention to the problem of linking the regions at higher levels of aggregation.
There are 146 countries in the ICP project and 155 basic headings. At this stage of the aggregation procedure, we assume that we have two 155 by 146 matrices of data: one matrix contains the PPPs for basic heading category n and country k, pnk, and the other
contains country expenditures in each country’s currency, enk, so that the notation is
basically the same as in the previous section but now k runs over all 146 countries instead of just the countries in a given region. At this stage, we could use any suitable multilateral method to aggregate up these data into a set of 146 country PPP’s and volumes, such as the EKS or IDB methods explained in the previous section. Call this
Approach 1. However, the problem with this approach is that the multilateral method to
be used would not necessarily respect the regional PPP’s unless it was restricted in some manner.
Thus we consider Approach 2, which will link the regions, while respecting the within region overall PPP’s that the regions deem best for their purposes.23 The first step is to reorganize the countries into 5 regions (we regard the OECD/Eurostat/CIS countries as forming one region). Consider region r which has C(r) countries in it. Let pnrc denote the
within region PPP for basic heading class n and country c in region r24 and let enrc denote
the corresponding expenditure in local currency. The total regional expenditure on commodity group n in currency units of country 1 in each region, Enr, is defined as
follows:
(31) Enr ≡ pnr1 ∑c=1C(r) enrc/pnrc ; r = 1,...,5 ; n = 1,...,155.
The corresponding regional PPPs by region and commodity, Pnr, are defined to be the
world BH parities for the numeraire country in each region:
(32) Pnr ≡ pnr1 ; r = 1,...,5 ; n = 1,...,155.
Now each region can be treated as if it were a single supercountry with supercountry expenditures Enr and basic heading PPPs Pnr defined by (31) and (32) respectively for the
5 supercountries and any of the linking methods described in the previous section can be used to link the regions. Once the interregional price and volumes have been determined, the regional price and volume aggregates can be used to provide world wide price and volume comparisons for each individual country. This method necessarily preserves all regional relative parities. Hill (2007e) attempted to show that the overall procedure does not depend on the choice of numeraire countries, either within regions or between regions; i.e., the relative country parities will be the same no matter what the choices are for the numeraire countries. However, Sergeev (2009) noted that this method is not
23 This approach was proposed by Diewert (2004b; 45-47). It is further described in much more detail by Hill (2007e).
24 The parities pnrc are the interregionally consistent PPP’s that were linked across regions as described in section 3 above; i.e., the pnrc are the estimated parameters ar brc cn which occur on the right hand side of equations (7). Assuming that country 1 is the numeraire country in each region, then the pnr1 are the parities that link the numeraire countries in each region.
invariant to the choice of the numeraire countries within the regions. Thus this method should probably not be used in the next round of the ICP in 2011.
Approach 2 in conjunction with the EKS method was used to link the regions in the current ICP round; i.e., the EKS method was used to link the 5 supercountry regions. Hill (2007e) discusses other possible methods that could be used to link the regions and these various alternative methods should part of the research agenda for the next round of comparisons. In particular, at higher levels of aggregation, we need to use the results of the present round to evaluate whether regional fixity is a good idea or not. The problem with regional fixity is that countries are not homogeneous within each region. In principle, it makes sense to compare countries whose (relative) price structures are similar and whose (absolute) quantity structures are similar: index number comparisons of price and volumes will work best under these conditions. Thus roughly speaking, it makes sense to compare directly countries who are at the same stage of development and build up a complete set of multilateral comparisons by linking (bilaterally) countries who are most structurally similar.25 R.J. Hill (1997) (1999a) (1999b) (2001) (2004) has developed methodology along these lines and it should be tested out using the detailed data generated by the present round.26 It may well be that the fixity methodology developed in this round is not the most appropriate methodology for subsequent rounds.
6. Problem Areas and the Future Research Agenda
There are a number of problem areas associated with making international comparisons that require additional research and discussion before the next round of the ICP takes place:
• If a country experiences hyperinflation during the reference year, the average price concept may not be meaningful. A possible solution to this problem is to use within the year inflation rates to “discount” prices collected throughout the year to a single reference week or day.27
• The problem of pricing exports and imports.28 At present, exchange rates are taken as the price of exports and imports. This is a reasonable approximation in some cases but the question is can we do anything better (that is not too costly)?
25 See Diewert (2002) for a discussion on how to measure structural similarity.
26 Another interesting issue is this: the present fixity imposed procedure is essentially a two stage GEKS procedure. At the first stage, countries are compared using GEKS within each region and then at the second stage, the five regions are linked together using another round of GEKS. Question: how does this two stage procedure compare to a single stage GEKS procedure using all 146 countries? The answer will probably be: they generate rather different parities. What then? What is the “truth”? We need criteria to determine “truth”. We could look at the axiomatic properties of two stage methods as compared to single stage methods but I do not believe that this would resolve the issues. At this point, I would fall back on the spatial linking methodology: it makes sense to build up a path of linked comparisons where we link together the countries which are most similar in structure.
27 See Hill (1996) for a discussion of the accounting problems when there is high inflation.
28 See Heston and Summers (2008; 4) for a discussion of this problem. A first approach to the problem would be to coordinate the calculation of national unit value export and import indexes across countries. This is a separate exercise that should be started well before the next ICP round. O’Connor (2008)
• The problem of negative expenditure categories. This problem arises with the net export category and the net additions to inventory category. Typically, there is not a problem provided that we do not attempt to provide PPPs for a single category that could be positive or negative across countries.29 If it is necessary to provide PPPs across countries for such a category, the problems can be avoided by providing separate PPPs for exports and imports or for starting and finishing inventory stocks and users can difference the results.
• Inaccurate expenditure weights can cause grave difficulties. In the next ICP round, it would be very desirable to have more accurate information on expenditures by basic heading available from participating countries.
• Methodological difficulties with hard to measure areas of the accounts. There are particular problems with the treatment of housing30, financial services and nonmarket production.31 These are problem areas for regular country accounts as well due to the lack of consensus on an appropriate methodology. Hopefully, international groups and academic economists interested in measurement problems will undertake additional research in these areas before the next ICP round.
• There is a very basic problem that makes international comparisons of prices and volumes very difficult and that is the lack of matching of products. The same problem occurs in the time series context due to the introduction of new products and the disappearance of “old” products but the lack of matching is much worse in the international context due to differences in tastes and big differences in the levels of development across countries, leading to very different consumption patterns. However, Structured Product Descriptions were introduced in the current ICP round and this does open up the possibility for undertaking hedonic regression exercises in the next round in order to improve the matching process. There are many problems to be addressed however,32 and it would be wise to undertake experimental hedonic studies well in advance of the next round.
• The fact that the ring list of commodities was somewhat different from the regional lists means that there is the possibility of anomalies in the final results; i.e., if different products are priced in the ring list, we cannot be sure the relative ring price levels really match up with the relative prices within the regions. The ring list of commodities was not determined completely independently from the mentions that the problems associated with calculating export and import price indexes is getting worse over time due to increasing trade in multinational intermediate goods and the transfer price problem. 29 Index number theory tends to break down if a value aggregate crosses zero or is equal to zero!
30 One area that we have not addressed is the impact of different procedures in different regions. For example, Asia and Africa used different methods for making productivity adjustments for government outputs index and they also used a different method for measuring housing output as compared to the CIS and South American regions. These problems need to be addressed well in advance of the next ICP round. 31 See Heston and Summers (2008), Giovannini (2008) and Bevacqua, Fantin, Quintslr and Ruiz (2008) for a discussion of these problems. The fact that current System of National Accounts conventions do not allow an imputed interest charge for capital that is used in the nonmarket sector tends to understate the contribution of this sector and the degree of understatement will not be constant across rich and poor countries.
32 See Hill and Timmer (2006) for a discussion of the problem of differing degrees of product overlap across countries.
country lists and this is all to the good.33 But in the next round, this integration of the ring product list with the regional product lists should be intensified with a best case scenario where the ring list becomes unnecessary.34
• It would be advisable to undertake some studies on alternative methods of aggregation at the higher levels of aggregation.35 In particular, the program of making comparisons based on the degree of similarity of the price and quantity data being compared that was initiated by Robert Hill (1999a) (1999b) (2001) (2004) seems to be sensible but users have not embraced it, perhaps due to the instability of the method.36 In any case, the World Bank now has a considerable data set based on the current ICP round that could be used to experiment with alternative methods of aggregation.
• Looking ahead into the more distant future, it would be desirable to integrate the ICP with the EU KLEMS project37, which is assembling data on the producer side of the economy as opposed to the final demand side, which is the focus of the ICP. Producer data are required in order to calculate relative productivity levels across economies, a topic of great interest to policy makers. Thus in addition to comparing components of final demand across countries, it would be desirable to
33 Yonas Biru who was responsible for organizing the ring list describes how this was done as follows: “The Global ring list was developed in close consultation with regional and country experts in an iterative processes. First, a consolidated global draft list was prepared that contained over 6,500 products from the five ICP regions and Eurostat-OECD comparison. Second, the list was then pruned by the Global Office to about 1,500 products, based on the country responses. The next step involved harmonizing product descriptions that originated from different regions and the list was sent back to the regions and a second round regional meetings were organized. A revised list was then created taking the second round comment from ring countries. The Global Office analyzed these second round country responses basic heading by basic heading to determine which products should be retained and which should be dropped. Key criteria for determining the final list included the number of regions and also the number of countries within each region where the product could be priced. A workshop was organized in Washington for regional coordinators and representatives of ring countries (one from each region) to go through the list and build consesnus on a global list. The workshop modified some products, dropped some and came up with the final list containing about 1200 products. In doing so the Global Office made sure that at least one product was represented from each region for each basic heading.” Other World Bank researchers who were involved with the ring project were Yuri Dikhanov, Ramgopal Erabelly, Nada Hamadeh, Farah Hussain, Jinsook Lee and Amy Lee.
34 Yonas Biru and Virgina Romand are currently undertaking a study to improve the SPD process. Yonas Biru describes the study as as follows: “We are developing coding structures not only for products but also for product characteristics for the next generation SPDs. This would facilitate mathing products across regions. This means we will be able to determine ring countries and ring products after data collection based on maximum overlap of products. Both the number and the mix of countries will be determined basic heading by basic heading. Potentially, linking can be done based on diferent criteria, including by ICP regions, consumption and price similarity indecies, etc. The method would also facilitate hedonic type regression, particularly for equipment goods.”
35 Alan Heston is currently undertaking such a study.
36 Robert Hill’s methodology for linking countries via a path of bilateral links for countries which have similar price and quantity structures has mainly been applied at higher levels of aggregation. Using a statistical approach, Hill and Timmer (2006) extend this similarity methodology to lower levels of aggregation where only price information is available and they also take into account situations where the amount of overlap in pricing products differs across countries. This is an important practical problem and their methods need to be studied and tested.
compare outputs and inputs by industry across countries so that international comparisons of sectoral productivity levels could be undertaken.38
7. Conclusion
My overall conclusion is that the 2005 ICP round was a big success. The regions liked the idea that they could define their own list of products for international pricing and this improved the quality of the data. The new methodology to link prices across the regions using ring countries also seems to be a clear improvement over previous rounds. Finally, the use of hand held computers and the structured product description methodology led to improvements in the production of national price statistics in many cases.39
One issue that has not been entirely satisfactorily resolved is the issue of disclosure of the data; i.e., a great deal of effort has gone in to collecting PPPs for 155 categories for 146 countries but only data on 15 highly aggregated PPPs will be released. Why the reluctance to release the data? Probably because at lower level of aggregation, the results can be quite unreliable. Still one would think that more than 15 categories could be released.40
As indicated in the previous section, some challenges remain but hopefully, these problems will be addressed before the next round takes place.
Appendix: The Properties of the Iklé Dikhanov Balk Multilateral System A.1 Introduction and Overview
Unfortunately, multilateral index number theory is much more complicated than bilateral index number theory. Thus a rather long appendix is required in order to investigate the axiomatic and economic properties of the IDB multilateral system, particularly when we allow some prices and quantities to be zero.41 A brief overview of this appendix follows. There are many equivalent ways of expressing the equations that define the IDB parities. Section 2 lists these alternative systems of equations that can be used to define the method. Section 3 provides proofs of the existence and uniqueness of solutions to the IDB equations. Section 4 considers various special cases of the IDB equations. When there are only two countries so that K = 2, we obtain a bilateral index number formula and this case is considered along with the case where N = 2, so that we have only 2 commodities. These special cases cast some light on the structure of the general indexes. Section 5 explores the axiomatic properties of the IDB method while section 6 looks at
38 See O’Connor (2008) for a similar long run proposal for the direction of the ICP. O’Connor also advocates making wealth comparisons across countries, which is feasible once we generate measures of capital input.
39 See Trewin (2008) and Fenwick and Whitestone (2008) on the externalities created by the ICP program. 40 Heston and Summers (2008; 5) and O’Connor (2008) discuss this issue.
41 Balk (1996; 207-208) has the most extensive published discussion of the properties of the IDB system but he considered only the case of positive prices and quantities for all commodities across all countries and he did not discuss the economic properties of the method.
the system’s economic properties. Finally, section 6 concludes by calculating a numerical example.
Throughout this appendix, we assume that the number of countries K and the number of commodities N is equal to or greater than two.
A.2 Alternative Representations A.2.1 The Pk, πn Representation
Recalling equations (15) and (16) in the main text, the basic data for the multilateral system are the prices and quantities for commodity n in country k at the basic heading level, pnk and ynk respectively, for n = 1,...,N and k = 1,...,K where the number of basic
heading categories N ≥ 2 and the number of countries K ≥ 2. The N by 1 vectors of prices and quantities for country k are denoted by pk and yk and their inner product is
pk⋅yk for k = 1,...,K. The share of country k expenditure on commodity n is denoted by snk ≡ pnkynk/pk⋅yk for k = 1,...,K and n = 1,...,N.
We will assume that for each n and k, either pnk, ynk and snk are all zero or pnk, ynk and snk
are all positive. Thus we allow for the possibility that some countries do not consume all of the basic heading commodities. This complicates the representations of the equations since division by zero prices, quantities or shares leads to difficulties and complicates proofs of existence.42 For now, we make the following assumptions:
(A1) For every basic heading commodity n, there exists a country k such that pnk, ynk and
snk are all positive so that each commodity is demanded by some country.
(A2) For every country k, there exists a commodity n such that pnk, ynk and snk are all
positive so that each country demands at least one basic heading commodity.
In section A.2, we will strengthen the above assumptions in order to ensure that the IDB equations have unique, positive solutions.
Recall that the IDB multilateral system was defined by the Dikhanov equations (27) and (28) (plus one normalization such as (23)). Taking into account the division by zero problem, these equations can be rewritten as follows:43
(A3) πn = [∑j=1K snj]/[∑k=1K (ynk Pk/pk⋅yk)] ; n = 1,...,N
(A4) Pk = pk⋅yk/π⋅yk ; k = 1,...,K where π is a vector whose components are π1,...,πN.
42 Balk’s (1996; 208) existence proof assumed that all prices and quantities were strictly positive.
43 Equations (A3) are equivalent to Balk’s (1996; 207) equations (38a) in the case where all price pnk are positive and equations (A4) are Balk’s equations (38b).
Using assumptions (A1) and (A2), it can be seen that equations (A3) and (A4) will be well behaved even if some pnk and ynk are zero. Equations (A3) and (A4) (plus a
normalization on the Pk or πn such as P1 = 1 or π1 = 1) provide our second representation
of the IDB multilateral equations.44
In order to find a solution to equations (A3) and (A4), one can start by assuming that π = 1N, a vector of ones and then use equations (A4) to determine a set of Pk. These Pk can
then be inserted into equations (A3) in order to determine a new π vector. Then this new π vector can be inserted into equations (A4) in order to determine a new set of Pk. And so on; the process can be continued until convergence is achieved.
A.2.2 An Alternative Pk, πn Representation using Biproportional Matrices
It can be seen that equations (A3) and (A4) can be rewritten in the following manner: (A5) ∑k=1K ynk [pk⋅yk]−1πnPk = ∑j=1K snj ; n = 1,...,N;
(A6) ∑n =1N ynk [pk⋅yk]−1πnPk = ∑n=1N snj = 1 ; k = 1,...,K.
Define the N by K normalized quantity matrix A which has element ank in row n and
column k where
(A7) ank ≡ ynk/pk⋅yk ; n = 1,...,N ; k = 1,...,K.
Define the N by K expenditure share matrix S which has the country k expenditure share for commodity n, snk in row n and column k. Let 1N and 1K be vectors of ones of
dimension N and K respectively. Then equations (A5) and (A6) can be written in matrix form as follows:45
(A8) AP = S1K ;
(A9) πTA = 1NTS
where π ≡ [π1,...,πN] is the vector of IDB international prices, P ≡ [P1,...,PK] is the vector
of DI country PPPs, denotes an N by N diagonal matrix with the elements of the vector π along the main diagonal and denotes an K by K diagonal matrix with the elements of the vector P along the main diagonal. There are N equations in (A8) and K equations in (A9). However, examining (A8) and (A9), it is evident that if N+K−1 of these equations are satisfied, then the remaining equation is also satisfied. Equations (A8) and (A9) are a special case of the biproportional matrix fitting model due to Deming and Stephan (1940) in the statistics context and to Stone (1962) in the economics context (the RAS method). Bacharach (1970; 45) studied this model in great detail and gave rigorous conditions for the existence of a unique positive π, P solution set to (A8), (A9)
44 Equations (27) and (28) provide a first representation in the case where all prices and quantities are positive.
45 Notation: when examining matrix equations, vectors such as π and P are to be regarded as column vectors and πT and PT denote their row vector transposes.
